Question

Read the following proof of the polynomial identity a3+b3=(a+b)3−3ab(a+b).

Step 1: a3+b3=a3+3a2b+3ab2+b3−3ab(a+b)
Step 2: a3+b3=a3+3a2b+3ab2+b3−3a2b−3ab2
Step 3: a3+b3=a3+3a2b−3a2b+3ab2−3ab2+b3
Step 4: a3+b3=a3+b3

Match each step with the correct reason.

Answer 1

given identity is

a³+b³ = (a+b)³ - 3ab(a+b)

we have to prove this

as we know that

(a+b)³= a³+b³+3a²b+3ab²

put this value in the identity

a³+b³ = a³+ b³+3a²b+3ab²-3ab(a+b)

now open the bracket

a³+b³= a³+b³+3a²b+3ab²-3a²b-3ab²

minus and will cancel each other

a³ + b³ = a³ + b³

hence prooved every step